Complex Base

The path corresponds to expressing complex integers X+i*Y in a base

    b = 5/2 + i*sqrt(3)/2

with coordinates scaled to put equilateral triangles on a square grid. So for integer X,Y on the triangular grid (X,Y either both odd or both even),

    X/2 + i*Y*sqrt(3)/2 = a[n]*b^n + ... + a[2]*b^2 + a[1]*b + a[0]

where each digit a[i] is either 0 or a sixth root of unity encoded into base-7 digits of N,

     w6 = e^(i*pi/3)            sixth root of unity, b = 2 + w6
        = 1/2 + i*sqrt(3)/2
     N digit     a[i] complex number
     -------     -------------------
       0          0
       1         w6^0 =  1
       2         w6^1 =  1/2 + i*sqrt(3)/2
       3         w6^2 = -1/2 + i*sqrt(3)/2
       4         w6^3 = -1
       5         w6^4 = -1/2 - i*sqrt(3)/2
       6         w6^5 =  1/2 - i*sqrt(3)/2

7 digits suffice because

     norm(b) = (5/2)^2 + (sqrt(3)/2)^2 = 7

Rotate Numbering

Parameter "numbering_type => 'rotate'" applies a rotation in each sub-part according to its location around the preceding level.

The effect can be illustrated by writing N in base-7. Part 10-16 is the same as the middle 0-6. Part 20-26 has a rotation by +60 degrees. Part 30-36 has rotation by +120 degrees, and so on.

                         22----21
                        /     /           numbering_type => 'rotate'
          31    36    23    20    26          N shown in base-7
         /  \     \     \        /
       32    30    35    24----25    13----12
         \        /                 /        \
          33----34     3---- 2    14    10----11
                     /        \     \
       46----45     4     0---- 1    15----16
               \     \
    41----40    44     5---- 6    64----63
      \        /                 /        \
       42----43    55----54    65    60    62
                  /        \     \     \  /
                56    50    53    66    61
                     /     /
                   51----52

Notice this means in each part the 11, 21, 31, etc, points are directed away from the middle in the same way, relative to the sub-part locations.

Working through the expansions gives the following rule for when an N is on the boundary of level k,

    write N in k many base-7 digits  (empty string if k=0)
    if any 0 digit then non-boundary
    ignore high digit and all 1 digits
    if any 4 or 5 digit then non-boundary
    if any 32, 33, 66 pair then non-boundary

A 0 digit is the middle of a block, or 4 or 5 digit the inner side of a block, for k>=1, hence non-boundary. After that the 6,1,2,3 parts variously expand with rotations so that a 66 is enclosed on the clockwise side and 32 and 33 on the anti-clockwise side.

Level Methods

"($n_lo, $n_hi) = $path->level_to_n_range($level)"

Return "(0, 7**$level - 1)".

Axis Rotations

In the fixed numbering, digit positions 1,2,3,4,5,6 go around +60deg each, so the N for rotation of X,Y by +60 degrees is each digit +1.

    N          = 0, 1, 2, 3, 4, 5, 6, 10, 11, 12
    rot+60(N)  = 0, 2, 3, 4, 5, 6, 1, 14, 16, 17, ... decimal
               = 0, 2, 3, 4, 5, 6, 1, 20, 22, 23, ... base7
    rot+120(N) = 0, 3, 4, 5, 6, 1, 2, 21, 24, 25, ... decimal
               = 0, 3, 4, 5, 6, 1, 2, 30, 33, 34, ... base7
    etc

In the rotate numbering, just adding +1 (etc) at the high digit alone is rotation.

X,Y Extents

The maximum X in a given level N=0 to 7^k-1 can be calculated from the replications. A given high digit 1 to 6 has sub-parts located at b^k*w6^(d-1). Those sub-parts are all the same, so the one with maximum real(b^k*w6^(d-1)) contains the maximum X.

    N_xmax_digit(j) = d=1to6 where real(w6^(d-1) * b^j) is maximum
                    = 1,1,6,6,6,5,5,5,4,4,4,3,3,3,3,2,2, ...
                 k-1
    N_xmax(k) = digits N_xmax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 8, 302, 2360, 16766, 100801, ...  decimal
              = 0, 1, 11, 611, 6611, 66611, 566611, ...  base7
                k-1
    z_xmax(k) = sum  w6^d[j] * b^j
                j=0      each d[j] with real(w6^d[j] * b^j) maximum
          = 0, 1, 7/2+1/2*sqrt3*i, 10-sqrt3*i, 57/2-3/2*sqrt3*i,...
    xmax(k) = 2*real(z_xmax(k))
            = 0, 2, 7, 20, 57, 151, 387, 1070, 2833, 7106, ...

For computer calculation these maximums can be calculated from the powers. The parts resulting can also be written in terms of the angle

    arg(b) = atan(sqrt(3)/5) = 19.106... degrees

For successive k, if adding this pushes the b^k angle past +30deg then the preceding digit goes past -30deg and becomes the new maximum X. Write the angle as a fraction of 60deg (pi/3),

    F = atan(sqrt(3)/5) / (pi/3)  = 0.318443 ...

This is irrational since b^k is never on the X or Y axes. That can be seen since 2/sqrt3*imag(b^k) mod 7 goes in a repeating pattern 1,5,4,6,2,3. Similarly 2*real(b^k) mod 7 so not on the Y axis, and also anything on the Y axis would have 3*k fall on the X axis.

Digits low to high are successive steps back cyclically 6,5,4,3,2,1 so that (with mod giving 0 to 5),

    N_xmax_digit(j) = (-floor(F*j+1/2) mod 6) + 1

The +1/2 is since initial direction b^0=1 is angle 0 which is half way between -30 and +30 deg.

Similarly for the location, using conj(w6) for rotation back

    z_xmax_exp(j) = floor(F*j+1/2)
                  = 0,0,1,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5, ...
    z_xmax(k) = sum(j=0,k-1, conj(w6)^z_xmax_exp(j) * b^j)

By symmetry the maximum extent is the same in 60deg, 120deg, etc directions, suitably rotated. The N in those cases has the digits 1,2,3,4,5,6 cycled around for the rotation. In PlanePath triangular X,Y coordinates direction 60deg means when sum X+3*Y is a maximum, etc.

If the +1/2 in the floor is omitted then the effect is to find the maximum point in direction +30deg. In the PlanePath coordinates this means maximum sum S = X+Y.

    N_smax_digit(j) = (-floor(F*j) mod 6) + 1
                    = 1,1,1,1,6,6,6,5,5,5,4,4,4,3,3, ...
                 k-1
    N_smax(k) = digits N_smax_digit(j)    low digit j=0
                 j=0
              = 0, 1, 8, 57, 400, 14806, 115648, ...     decimal
              = 0, 1, 11, 111, 1111, 61111, 661111, ...  base7
    and also N_smax() + 1
    z_smax_exp(j) = floor(F*j)
                  = 0,0,0,0,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6, ...
    z_smax(k) = sum(j=0,k-1, conj(w6)^z_smax_exp(j) * b^j)
              = 0, 1, 7/2+1/2*sqrt3*i, 9+3*sqrt3*i, 19+12*sqrt3*i, ...
    and also z_smax() + w6^2
    smax(k) = 2*real(z_smax(k)) + imag(z_smax(k))*2/sqrt3
            = 0, 2, 8, 24, 62, 172, 470, 1190, 3202, 8740, ...
              coordinate sum X+Y max

In the base figure, points 1 and 2 have the same X+Y=2 and this remains so in subsequent levels, so that for k>=1 N_smax(k) and N_smax(k)+1 are equal maximums.